Found problems: 85335
2008 ITest, 44
Now Wendy wanders over and joins Dr. Lisi and her younger siblings. Thinking she knows everything there is about how to work with arithmetic series, she nearly turns right around to walk back home when Dr. Lisi poses a more challenging problem. "Suppose I select two distinct terms at random from the $2008$ term sequence. What's the probability that their product is positive?" If $a$ and $b$ are relatively prime positive integers such that $a/b$ is the probability that the product of the two terms is positive, find the value of $a+b$.
1976 All Soviet Union Mathematical Olympiad, 225
Given $4$ vectors $a,b,c,d$ in the plane, such that $a+b+c+d=0$. Prove the following inequality: $$|a|+|b|+|c|+|d| \ge |a+d|+|b+d|+|c+d|$$
1999 French Mathematical Olympiad, Problem 3
For which acute-angled triangles is the ratio of the smallest side to the inradius the maximum?
2009 Today's Calculation Of Integral, 402
Consider a right circular cylinder with radius $ r$ of the base, hight $ h$. Find the volume of the solid by revolving the cylinder about a diameter of the base.
2009 Federal Competition For Advanced Students, P2, 4
Let $ a$ be a positive integer. Consider the sequence $ (a_n)$ defined as $ a_0\equal{}a$
and $ a_n\equal{}a_{n\minus{}1}\plus{}40^{n!}$ for $ n > 0$. Prove that the sequence $ (a_n)$ has infinitely
many numbers divisible by $ 2009$.
2018 Korea Winter Program Practice Test, 3
Let $n$ be a "Good Number" if sum of all divisors of $n$ is less than $2n$ for $n\in \mathbb{Z}.$
Does there exist an infinite set $M$ that satisfies the following?
For all $a,b\in M,$ $a+b$ is good number. ($a=b$ is allowed.)
2016 Dutch IMO TST, 1
Let $\triangle ABC$ be a acute triangle. Let $H$ the foot of the C-altitude in $AB$ such that $AH=3BH$, let $M$ and $N$ the midpoints of $AB$ and $AC$ and let $P$ be a point such that $NP=NC$ and $CP=CB$ and $B$, $P$ are located on different sides of the line $AC$. Prove that $\measuredangle APM=\measuredangle PBA$.
2011 Albania Team Selection Test, 2
The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.
2014 Bosnia And Herzegovina - Regional Olympiad, 2
Solve the equation $$x^2+y^2+z^2=686$$ where $x$, $y$ and $z$ are positive integers
2017 ASDAN Math Tournament, 6
Let $\triangle ABC$ be a right triangle with right angle $\angle B$. Suppose the angle bisector $l$ of $B$ divides the hypotenuse $AC$ into two segments of length $\sqrt{3}-1$ and $\sqrt{3}+1$. What is the measure of the smaller angle between $l$ and $AC$, in radians?
2024/2025 TOURNAMENT OF TOWNS, P3
A positive integer $M$ has been represented as a product of primes. Each of these primes is increased by 1 . The product $N$ of the new multipliers is divisible by $M$ . Prove that if we represent $N$ as a product of primes and increase each of them by 1 then the product of the new multipliers will be divisible by $N$ .
Alexandr Gribalko
2000 Moldova Team Selection Test, 5
Let $(F_n)_{n\in\mathbb{N}}$ be the Fibonacci sequence difined as $F_0=F_1=1, F_{n+2}=F_{n+1}+F_n, \forall n\in\mathbb{N}$. Show that for every nonnegative integer $r$ there is a term in the Fibonacci sequence that is divided by $r$.
2018 Iran Team Selection Test, 4
Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of $3$ and powers of $5$.
Prove that there exist infinitely many positive integers which they are not "useful but not optimized".
(e.g. $37=(3^0+3^1+3^3)+(5^0+5^1)$ is a " useful but not optimized" number)
[i]Proposed by Mohsen Jamali[/i]
2021 Science ON grade XII, 2
Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\
$\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\
$\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\
$\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true?
\\ \\
[i] (Bogdan Blaga)[/i]
2015 ASDAN Math Tournament, 10
Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other $2$ players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once?
2022 Bulgarian Spring Math Competition, Problem 12.4
Let $m$ and $n$ be positive integers and $p$ be a prime number. Find the greatest positive integer $s$ (as a function of $m,n$ and $p$) such that from a random set of $mnp$ positive integers we can choose $snp$ numbers, such that they can be partitioned into $s$ sets of $np$ numbers, such that the sum of the numbers in every group gives the same remainder when divided by $p$.
2002 Junior Balkan Team Selection Tests - Moldova, 10
The circles $C_1$ and $C_2$ intersect at the distinct points $M$ and $N$. Points $A$ and $B$ belong respectively to the circles $C_1$ and $C_2$ so that the chords $[MA]$ and $[MB]$ are tangent at point $M$ to the circles $C_2$ and $C_1$, respectively. To prove it that the angles $\angle MNA$ and $\angle MNB$ are equal.
2005 MOP Homework, 1
Two rooks on a chessboard are said to be attacking each other if they are placed in the same row or column of the board.
(a) There are eight rooks on a chessboard, none of them attacks any other. Prove that there is an even number of rooks on black fields.
(b) How many ways can eight mutually non-attacking rooks be placed on the 9 £ 9 chessboard so that all eight rooks are on squares of the same color.
2017 Azerbaijan Team Selection Test, 2
Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that
[list]
[*]$m = 1$ and $l = 2k$; or
[*]$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$.
[/list]
2022 Germany Team Selection Test, 3
Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.
2021 USAMTS Problems, 2
Sydney the squirrel is at $(0, 0)$ and is trying to get to $(2021, 2022)$. She can move only by reflecting her position over any line that can be formed by connecting two lattice points, provided that the reflection puts her on another lattice point. Is it possible for Sydney to reach $(2021, 2022)$?
2024 IFYM, Sozopol, 8
Each cell in a \( 2024 \times 2024 \) table contains the letter \( A \) or \( B \), with the number of \( A \)'s in each row being the same and the number of \( B \)'s in each column being the same. Alexandra and Boris play the following game, alternating turns, with Alexandra going first. On each turn, the player chooses a row or column and erases all the letters in it that have not yet been erased, as long as at least one letter is erased during the turn, and at the end of the turn, at least one letter remains in the table. The game ends when exactly one letter remains in the table. Alexandra wins the game if the letter is \( A \), and Boris wins if it is \( B \). What is the number of initial tables for which Alexandra has a winning strategy?
1986 AMC 12/AHSME, 19
A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?
$ \textbf{(A)}\ \sqrt{13}\qquad\textbf{(B)}\ \sqrt{14}\qquad\textbf{(C)}\ \sqrt{15}\qquad\textbf{(D)}\ \sqrt{16}\qquad\textbf{(E)}\ \sqrt{17}$
2023 LMT Fall, 7
Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$.
[i]Proposed by Isabella Li[/i]
2022 AMC 8 -, 15
Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?
[asy]
//diagram by pog
size(8.5cm);
usepackage("mathptmx");
defaultpen(mediumgray*0.5+gray*0.5+linewidth(0.63));
add(grid(6,6));
label(scale(0.7)*"$1$", (1,-0.3), black);
label(scale(0.7)*"$2$", (2,-0.3), black);
label(scale(0.7)*"$3$", (3,-0.3), black);
label(scale(0.7)*"$4$", (4,-0.3), black);
label(scale(0.7)*"$5$", (5,-0.3), black);
label(scale(0.7)*"$1$", (-0.3,1), black);
label(scale(0.7)*"$2$", (-0.3,2), black);
label(scale(0.7)*"$3$", (-0.3,3), black);
label(scale(0.7)*"$4$", (-0.3,4), black);
label(scale(0.7)*"$5$", (-0.3,5), black);
label(scale(0.75)*rotate(90)*"Price (dollars)", (-1,3.2), black);
label(scale(0.75)*"Weight (ounces)", (3.2,-1), black);
dot((1,1.2),black);
dot((1,1.7),black);
dot((1,2),black);
dot((1,2.8),black);
dot((1.5,2.1),black);
dot((1.5,3),black);
dot((1.5,3.3),black);
dot((1.5,3.75),black);
dot((2,2),black);
dot((2,2.9),black);
dot((2,3),black);
dot((2,4),black);
dot((2,4.35),black);
dot((2,4.8),black);
dot((2.5,2.7),black);
dot((2.5,3.7),black);
dot((2.5,4.2),black);
dot((2.5,4.4),black);
dot((3,2.5),black);
dot((3,3.4),black);
dot((3,4.2),black);
dot((3.5,3.8),black);
dot((3.5,4.5),black);
dot((3.5,4.8),black);
dot((4,3.9),black);
dot((4,5.1),black);
dot((4.5,4.75),black);
dot((4.5,5),black);
dot((5,4.5),black);
dot((5,5),black);
[/asy]
$\textbf{(A)} ~1\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~3\qquad\textbf{(D)} ~4\qquad\textbf{(E)} ~5\qquad$